# Functional Regression Control Chart

In this vignette we show how to use the funcharts package to apply the methods proposed in Centofanti et al. (2020) to build the functional regression control chart, when we have a functional response variable and multivariate functional covariates. Let us show how the funcharts package works through an example with the dataset air, which has been included from the R package FRegSigCom and is used in the paper of Qi and Luo (2019). The authors propose a function-on-function regression model of the NO2 functional variable on all the other functional variables available in the dataset.

NOTE: since the objective of this vignette is only to illustrate how the package works, in the following we will use only 15 basis functions and a fixed smoothing parameter to reduce the computational time.

First of all, starting from the discrete data, let us build the multivariate functional data objects of class mfd, see vignette("mfd").

library(funcharts)
data("air")
fun_covariates <- names(air)[names(air) != "NO2"]
mfdobj <- get_mfd_list(air,
grid = 1:24,
n_basis = 15,
lambda = 1e-2)
mfdobj_y <- mfdobj[, "NO2"]
mfdobj_x <- mfdobj[, fun_covariates]

In order to perform the statistical process monitoring analysis, we divide the data set into a phase I and a phase II dataset.

rows1 <- 1:300
rows2 <- 301:355
mfdobj_x1 <- mfdobj_x[rows1]
mfdobj_x2 <- mfdobj_x[rows2]
mfdobj_y1 <- mfdobj_y[rows1]
mfdobj_y2 <- mfdobj_y[rows2]

## Function-on-function regression based on principal components

We can build a function-on-function linear regression model using the function fof_pc, which is based on the multivariate functional principal component analysis (MFPCA) on the multivariate functional covariates and the functional response.

The components to retain in the MFPCA models are selected according to the total variance explained criterion, which can be set in the arguments tot_variance_explained_x, tot_variance_explained_y and tot_variance_explained_res (for the residuals). One can also set the type_residuals in order to choose between standard residuals and studentized residuals (see Centofanti et al. (2020)).

mod_fof <- fof_pc(mfdobj_y = mfdobj_y1, mfdobj_x = mfdobj_x1)

As a result you get a list with the original data used for model estimation, the result of applying pca_mfd on the multivariate functional covariates, the functional response, and the residuals, the estimated regression model and additional information. It is possible to plot the estimated functional regression coefficients, which is a bivariate functional data object of class bifd using the function plot_bifd:

plot_bifd(mod_fof$beta_fd) ## Plotting the functional regression control chart The function regr_cc_fof provides a data frame with all the information required to plot the desired functional regression control chart (see Centofanti et al. (2020)). Among the arguments, you can pass a tuning data set with the arguments mfdobj_y_tuning and mfdobj_x_tuning, which is not used for model estimation/training, but is used only to estimate control chart limits. If this data set is not provided, control chart limits are calculated on the basis of the training data. The arguments mfdobj_y_new and mfdobj_x_new contain the phase II data set of observations of the functional response and multivariate functional covariates that are to be monitored with the control charts. The function plot_control_charts returns a ggplot with the control charts. frcc_df <- regr_cc_fof(object = mod_fof, mfdobj_y_new = mfdobj_y2, mfdobj_x_new = mfdobj_x2) plot_control_charts(frcc_df) We can also plot the new functions to be monitored, against the reference training data set, by first using predict_fof_pc, which produces the prediction of new observations of the functional response given the new observations of the functional covariates, as well as the corresponding prediction error: y_hat <- predict_fof_pc(object = mod_fof, mfdobj_y_new = mfdobj_y2, mfdobj_x_new = mfdobj_x2) and then using plot_mon to plot a given observation against the reference data set of prediction errors/residuals used to estimate the control chart limits. Here for example we consider the observation 54 plot_mon(cclist = frcc_df, fd_train = mod_fof$residuals,
fd_test = y_hat\$pred_error[54])

## Real-time functional regression control chart

As for the scalar-on-function case shown in Capezza et al. (2020), we also provide the real-time version of the functional regression control chart. Given the domain interval $$(a, b)$$ of the functional data, for each current domain point $$k$$ to be monitored, it filters the available functional data in the interval $$(a,k)$$. The function get_mfd_list_real_time gives a list of functional data objects each evolving up to the intermediate domain point $$k$$.

mfd_list <- get_mfd_list_real_time(data_list = air,
grid = 1:24,
n_basis = 15,
lambda = 1e-2,
k_seq = seq(0.5, 1, length.out = 7))

mfd_list_x1 <- lapply(mfd_list, function(x) x[rows1, fun_covariates])
mfd_list_x2 <- lapply(mfd_list, function(x) x[rows2, fun_covariates])
mfd_list_y1 <- lapply(mfd_list, function(x) x[rows1, "NO2"])
mfd_list_y2 <- lapply(mfd_list, function(x) x[rows2, "NO2"])

Then, the function fof_pc_real_time applies the function fof_pc to each element in mfd_list_x1 and mfd_list_x2.

mod_fof_pc_real_time_list <- fof_pc_real_time(
mfdobj_y_list = mfd_list_y1,
mfdobj_x_list = mfd_list_x1)

Then, we can use control_charts_fof_pc_real_time to apply control_charts_fof_pc to each element in mod_fof_pc_real_time_list and produce control charts for the phase II data mfd_list_y2 and mfd_list_x2.

cc_list_real_time <- regr_cc_fof_real_time(
mod_list = mod_fof_pc_real_time_list,
mfdobj_y_new_list = mfd_list_y2,
mfdobj_x_new_list = mfd_list_x2
)

Finally, we can plot the real-time control charts for a single observations, giving for each $$k$$ the monitoring statistics calculated on the data observed in $$(a, k)$$. Here follows an example showing the real time control charts for a single phase II observation (id number 54).

plot_control_charts_real_time(cc_list_real_time, id_num = 54)
#> Warning: guides(<scale> = FALSE) is deprecated. Please use guides(<scale> = "none") instead.

#> Warning: guides(<scale> = FALSE) is deprecated. Please use guides(<scale> = "none") instead.

# References

• Capezza C, Lepore A, Menafoglio A, Palumbo B, Vantini S. (2020) Control charts for monitoring ship operating conditions and CO2 emissions based on scalar-on-function regression. Applied Stochastic Models in Business and Industry, 36(3):477–500. https://doi.org/10.1002/asmb.2507
• Centofanti F, Lepore A, Menafoglio A, Palumbo B, Vantini S. (2020) Functional Regression Control Chart. Technometrics. https://doi.org/10.1080/00401706.2020.1753581
• Qi X, Luo R. (2019). Nonlinear function-on-function additive model with multiple predictor curves. Statistica Sinica, 29:719–739. https://doi.org/10.5705/ss.202017.0249